Electrical – To find the maximum power that can be delivered to resistor that is specified

Barry gives the correct answer. What follows is the mathematical justification.

The power delivered to the load resistor, for a given Thevenin equivalent circuit is:

\$P_L = V_L \cdot I_L = V_{th}\dfrac{R_L}{R_{th}+ R_L} \cdot \dfrac{V_{th}}{R_{th}+ R_L} = V^2_{th} \dfrac{R_L}{(R_{th}+ R_L)^2}\$

If we fix \$R_{th}\$ and ask which value for \$R_L\$ gives maximum \$P_L\$, the value is given by \$R_L = R_{th}\$ and the resulting power delivered to the load is:

\$P_{L,max} = \dfrac{V^2_{th}}{4R_{th}} \$

However, if we fix \$R_L \$, and ask which value for \$R_{th}\$ gives maximum \$P_L\$, the answer is, by inspection, \$R_{th} = 0\$.

Thus, the answer is \$R = 500 \Omega \$ so that \$R_{th} = 0\$

Those statements are true. But I see what bothers you. You have to imagine it like this:

In Wireless transmission one wants to make the highest loss possible since this loss will actually be the power of the transmitted signal. This is a very easy optimization problem which will lead to the fact you have stated $$R_L = R_{in}.$$

In power transmission however the goal is to be as efficient as possible. This will still lead to the fact that the highest power output is achieved with $$R_L = R_{in}.$$ But thats the maximum you could get out e.g. of a transformer. You don't want it to operate in this point. Instead you lay out your transformer in such a way that it won't reach this point. Then the efficiency is much higher but you won't be using its full potential.

You might have noticed this when e.g. trying to charge a tablet with phone charger. It may works and the charger maxes out, but it gets hot since much energy is converted to heat because of the high internal loss. (Keep in mind that a phone charger is not only a transformer). You take a bigger charger which won't max out, is may oversized but has a higher efficiency.

The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor.

This is also good explained in Wikipedia:

In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer.